Method for disturbance compensation based on sliding mode disturbance observer for spacecraft with large flexible appendage

ABSTRACT

The present invention provides a method for disturbance compensation based on a sliding mode disturbance observer for a spacecraft with a large flexible appendage, comprising steps of: a) building a spacecraft attitude control system; b) constructing an external system, the external system being incorporated with an uncertain portion of a damping matrix of a flexible appendage of the spacecraft; the external system being incorporated with an uncertain portion of a rigidity matrix of the flexible appendage of the spacecraft and describing a sum of flexible vibration and environmental disturbance; c) configuring a sliding mode disturbance observer for estimating the value of the sum of flexible vibration and environmental disturbance; d) compounding a nominal controller with the sliding mode disturbance observer in step c) to obtain a compound controller; the compound controller compensating for the sum of flexible vibration and environmental disturbance.

CROSS-REFERENCE TO RELATED APPLICATIONS

This non-provisional application claims priority and benefit of, under35 U.S.C. §119(a), Patent Application No. 201610361655.0 filed in P.R.China on May 26, 2016, the entire content of which is herebyincorporated by reference.

FIELD OF THE INVENTION

The present invention relates to the field of aerospace technology, andin particular, to a method for disturbance compensation based on asliding mode disturbance observer for a spacecraft with a large flexibleappendage.

BACKGROUND OF THE INVENTION

With the development of aerospace technology, the attitude controltechnology with high accuracy has become a critical technology forcertain spacecrafts to realize their functions, for example,communication satellites, remote sensing satellites, space telescopes,etc. Therefore, the issue of high-accuracy attitude control of aspacecraft has become a hot and difficult point in researches recently.As for an inflexible satellite, its kinetic model is relatively simple,and the corresponding control method is relatively mature. However, inrecent years, a flexible spacecraft, especially a spacecraft with alarge appendage has become an important direction for the development ofaerospace technology in the future, for example the ETS-VIII satellitein Japan. These large flexible satellites carry light flexibleappendages such as large expandable antenna arrays and solar panels, andthe kinetic model of the spacecraft becomes very complicated since it isa typical non-linear, multi-coupled distribution parameter system withinfinite degree of freedom. This brings along a great challenge to thehigh-accuracy attitude control of the spacecraft. In addition, theselarge flexible appendages tend to produce elastic vibration and flexibleappendages with large area increase the influence of environmentaldisturbances such as aerodynamic drag, solar radiation pressure and thelike. These vibrations and external disturbances further increase thedifficulty of attitude control of the spacecraft. Accordingly, amanti-disturbance control method with a high accuracy becomes abottleneck technology for attitude control of a large flexiblespacecraft.

As for the problem of attitude control of a flexible spacecraft,different control methods are also proposed in order to offset orrestrain the influence of flexible vibration and external disturbance,and the typical ones include H_(∞) control, self-adaptation control,sliding mode variable structure control and so on. However, most ofthese control methods do not have typical disturbance offset ability,and thus cause the control accuracy to be limited. Based on the kineticmodel of the system, flexible vibration and environmental disturbancecan be described with referenced to the external system. However, due tothe error of measurement and the difference between the spaceenvironment and ground environment, the damping and frequency parametersof the flexible appendage measured in the ground experiments always havea big uncertainty, which causes the external system for describing thedisturbance to be a mathematical model with uncertain parameters.Internal mode control, active disturbance rejection control (ADRC) anddisturbance observer-based control (DOBC) are relatively typicaldisturbance compensation methods. However, the traditional internal modecontrol has a high requirement on the disturbance model, and requiresthe disturbance model to be accurately known. ADRC estimates disturbanceby way of an expansion state observer without the use of the intrinsicinformation of the disturbance itself, and thus has certainconservatism. DOBC makes full use of the information of disturbance,estimates and compensates for the disturbance that can be modeled in thedisturbance system, and achieves an ideal effect, and it can also allowthe disturbance model to have certain uncertainty. However, the rate ofconvergence of observation error of the traditional DOBC cannot beguaranteed, and gain scheduling is very complicated, while a slidingmode observer is superior in being not sensitive to parameter change anddisturbance and has a high rate of convergence. Therefore, theadvantages of the traditional DOBC and the sliding mode observer can becombined by using a sliding mode disturbance observer for estimatingdisturbance. In this way, the disturbance model is utilized, meanwhilethis method has a strong robustness to the change of the disturbancemodel. Furthermore, gain scheduling is very easy, and it can be ensuredthat the observation error can be converged into a certain adjustablearea rapidly, thereby improving the accuracy, robustness and rapidity ofdisturbance estimation.

Therefore, there is a need for a method for disturbance compensationbased on a sliding mode disturbance observer for a spacecraft with alarge flexible appendage that can efficiently estimate flexiblevibration and environmental disturbance.

SUMMARY OF THE INVENTION

An objective of the present invention is to The present inventionprovides a sliding mode disturbance observer with high observationaccuracy, strong robustness and easy gain scheduling, which solves thedifficulty of accurate estimation and compensation for disturbance withuncertain parameters that can be modeled, and improves the controlaccuracy of the system.

According to one aspect of the present invention, a method fordisturbance compensation based on a sliding mode disturbance observerfor a spacecraft with a large flexible appendage is provided, comprisingthe following steps of:

a) building a spacecraft attitude control system Σ₁, the spacecraftattitude control system Σ₁ being incorporated with environmentaldisturbance and being converted into the spacecraft attitude controlsystem Σ₂, and the spacecraft attitude control system Σ₂ beingincorporated with a sum of flexible vibration and environmentaldisturbance;

b) constructing an external system Σ₃, the external system Σ₃ describingthe sum of flexible vibration and environmental disturbance;

wherein, the external system Σ₃ is constructed through the followingsteps:

(1) incorporating an uncertain portion C_(Δ) of a damping matrix C ofthe flexible appendage of the spacecraft, and incorporating an uncertainportion D_(Δ) of a rigidity matrix D of the flexible appendage of thespacecraft; describing the damping matrix and rigidity matrix of thespacecraft as below:

$\quad\left\{ \begin{matrix}{C = {C_{0} + C_{\Delta}}} \\{D = {D_{0} + D_{\Delta}}}\end{matrix} \right.$

in which, C₀ and D₀ are respectively nominal parameters measured on theground;

(2) defining state variables w₁=η, w₂={dot over (η)} and w₃=d, obtainingthe following equation:

$\begin{bmatrix}{\overset{.}{w}}_{1} \\{\overset{.}{w}}_{2} \\{\overset{.}{w}}_{3}\end{bmatrix} = {{\begin{bmatrix}0 & I & 0 \\{- {GD}} & {- {GC}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}\begin{bmatrix}w_{1} \\w_{2} \\w_{3}\end{bmatrix}} - {\begin{bmatrix}0 \\G \\0\end{bmatrix}\delta\;{J^{- 1}\left( {{{- \omega^{x}}J\;\omega} + u} \right)}} + \begin{bmatrix}0 \\0 \\\overset{.}{d}\end{bmatrix}}$

in which, I is a unit matrix, and a matrix G=(I−δJ⁻¹δ^(T))⁻¹;

(3) defining the following coefficient matrix:

${W = \begin{bmatrix}0 & I & 0 \\{- {GD}_{0}} & {- {GC}_{0}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}},{B = {{- \begin{bmatrix}0 \\G \\0\end{bmatrix}}\delta\; J^{- 1}}},{{{{and}\mspace{14mu} V} = \left\lbrack {\delta^{T}D\mspace{14mu}\delta^{T}C\mspace{14mu} I} \right\rbrack};}$

(4) the external system Σ₃ being described as below:

$\sum\limits_{3}{:\left\{ \begin{matrix}{\overset{.}{w} = {{\left( {W + W_{\Delta}} \right)w} + {B\left( {{{- \omega^{x}}J\;\omega} + u} \right)} + \Gamma}} \\{\overset{\_}{d} = {V\; w}}\end{matrix} \right.}$

in which, w=[w₁ ^(T) w₂ ^(T) w₃ ^(T)]^(T), Γ is an uncertain vector, andΓ is expressed as: Γ=[0 0 {dot over (d)}]^(T); W_(Δ) satisfies a boundedcondition W_(Δ)=MF(t)N, M and N are constant matrixes of a proper numberof dimensions, F(t) is a time-varying matrix and satisfiesF^(T)(t)F(t)≦I; a state variable w satisfies a norm bounded condition∥w∥≦α, the sum d of flexible vibration and environmental disturbancesatisfies a norm bounded condition ∥d∥≦β, in which α and β are knownconstants.

c) configuring a sliding mode disturbance observer for estimating thevalue of the sum of flexible vibration and environmental disturbance;

d) compounding a nominal controller with the sliding mode disturbanceobserver in step c) to obtain a compound controller;

the compound controller compensating for the sum of flexible vibrationand environmental disturbance according to the estimated value of thesum of flexible vibration and environmental disturbance.

In certain aspects, the present invention relates to a spacecraft usingthe method as described above.

According to another aspect of the present invention, a spacecraft witha large flexible appendage based on a sliding mode disturbance observeris provided, which comprises a spacecraft shell, an external systemmodule, a sliding mode disturbance observation module, a nominal controlmodule, a compound control module, a central processing unit (CPU), acontrol unit and a spacecraft flexible wing plate and a spacecraftattitude control module, wherein,

the spacecraft flexible wing plate is unfolded at two ends of thespacecraft shell;

the external system module is configured to describe the sum of flexiblevibration and external environmental disturbance by the external system,and deliver the description result of the sum of flexible vibration andexternal environmental disturbance to the compound control module;

the sliding mode disturbance observation module is configured toestimate the sum of flexible vibration and external environmentaldisturbance by a sliding mode disturbance observer;

the nominal control module is configured to control a nominal controllerto compound with the sliding mode disturbance observer in the slidingmode disturbance observation module;

the compound control module is configured to compensate for the sum offlexible vibration and external environmental disturbance according tothe estimated value {circumflex over (d)} of the sum d of flexiblevibration and external environmental disturbance by a compoundcontroller;

the spacecraft attitude control module is configured to incorporate thesum of flexible vibration and external environmental disturbance;

the central processing unit (CPU) reads the data of the compound controlmodule 205, and processes the data; and

the control unit executes the processing result of the centralprocessing unit (CPU) and controls the attitude of the spacecraft.

It shall be appreciated that both the aforesaid general description andthe following specific description are intended for illustrativedescription and explanation, and shall not be construed as for limitingthe contents to be protected in the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objectives, effects, and advantages of the present inventionwill become apparent from the following description of the embodimentsof the present invention with reference to the accompanying drawings,wherein:

FIG. 1 illustrates a design flow chart of the method for disturbancecompensation based on a sliding mode disturbance observer for aspacecraft with a large flexible appendage according to the presentinvention;

FIG. 2 shows a module block diagram of a spacecraft according to anembodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

Objects and functions of the present invention as well as methods forrealizing these objects and functions will be elucidated with referenceto exemplary embodiments. However, the present invention is not limitedto the following disclosed exemplary embodiments, but may be implementedin different ways. The description of the invention is merely providedto assist those of ordinary skill in the art in a comprehensiveunderstanding of specific details of the invention in nature.

As used herein, the term “module” may refer to, be part of, or includean Application Specific Integrated Circuit (ASIC); an electroniccircuit; a combinational logic circuit; a field programmable gate array(FPGA); a processor (shared, dedicated, or group) that executes code;other suitable hardware components that provide the describedfunctionality; or a combination of some or all of the above, such as ina system-on-chip. The term module may include memory (shared, dedicated,or group) that stores code executed by the processor.

Hereinafter, embodiments of the present invention will be described withreference to the drawings. In the drawings, like reference numeralsdesignate like or similar parts or steps.

The present invention provides a method for disturbance compensationbased on a sliding mode disturbance observer for a spacecraft with alarge flexible appendage. As shown in FIG. 1, it is a design flow chartof the method for disturbance compensation based on a sliding modedisturbance observer for a spacecraft with a large flexible appendageaccording to the present invention. In the spacecraft disturbance method100 of this embodiment, a spacecraft attitude control system is built,and an external system is constructed for describing the sum of flexiblevibration and external environmental disturbance of the spacecraft. Asliding mode disturbance observer is configured for estimating the valueof the sum of flexible vibration and external environmental disturbance,and the sliding mode disturbance observer is compounded with a nominalcontroller to compensate for the sum of flexible vibration and externalenvironmental disturbance and stabilize the attitude control system.

With the purpose of illustration, the method for disturbancecompensation based on a sliding mode disturbance observer for aspacecraft with a large flexible appendage provided by the presentinvention is implemented through different modules. As shown in FIG. 2,it is a module block diagram of the spacecraft with a large flexibleappendage based on a sliding mode disturbance observer according to anembodiment of the present invention. Specifically, the spacecraftcomprises a spacecraft shell 201, an external system module 202, asliding mode disturbance observation module 203, a nominal controlmodule 204, a compound control module 205, a central processing unit(CPU) 206, a control unit 207, a spacecraft flexible wing plate 208 anda spacecraft attitude control module 209.

As shown in FIG. 2, the sliding mode disturbance observation module 203,the nominal control module 204, the compound control module 205, thecentral processing unit (CPU) 206, the control unit 207, the externalsystem module 202 and the spacecraft attitude control module 209 aremounted inside the spacecraft shell 201. The spacecraft flexible wingplate 208 is unfolded at two ends of the spacecraft shell 201.

The external system module 202 is configured to describe the sum offlexible vibration and external environmental disturbance by theexternal system. The external system module 202 delivers the descriptionresult of the sum of flexible vibration and external environmentaldisturbance to the compound control module 205.

The sliding mode disturbance observation module 203 is configured toestimate the sum of flexible vibration and external environmentaldisturbance by a sliding mode disturbance observer.

The nominal control module 204 is configured to control a nominalcontroller to compound with the sliding mode disturbance observer in thesliding mode disturbance observation module 203.

The compound control module 205 is configured to compensate for the sumof flexible vibration and external environmental disturbance accordingto the estimated value {circumflex over (d)} of the sum d of flexiblevibration and external environmental disturbance by a compoundcontroller.

The spacecraft attitude control module 209 is configured to incorporatethe sum of flexible vibration and external environmental disturbance.

The central processing unit (CPU) 206 reads the data of the compoundcontrol module 205, and processes the data.

The control unit 207 executes the processing result of the centralprocessing unit (CPU) 206 and controls the attitude of the spacecraft.Specifically, the control unit 207 compensates for the sum of flexiblevibration and external environmental disturbance according to theestimated value {circumflex over (d)} of the sum d of flexible vibrationand external environmental disturbance through the compound controlmodule 205, there by adjusting the attitude of the spacecraft. Themethod for disturbance compensation based on a sliding mode disturbanceobserver for a spacecraft with a large flexible appendage will bedescribed in detail with reference to FIG. 1. The specific steps are asfollows:

In step S101, building a spacecraft attitude control system, and the sumof flexible vibration and external environmental disturbance isincorporated into the spacecraft attitude control system.

An external environmental disturbance is incorporated, and a spacecraftattitude control system Σ₁ is built. It is expressed as:

$\sum\limits_{1}{:\left\{ \begin{matrix}{{{J\;\overset{.}{\omega}} + {\delta^{T}\overset{¨}{\eta}}} = {{{- \omega^{x}}J\;\omega} + u + d}} \\{{\overset{¨}{\eta} + {C\;\overset{.}{\eta}} + {D\;\eta} + {\delta\;\overset{.}{\omega}}} = 0}\end{matrix} \right.}$in which, J is an inertia matrix of the flexible spacecraft, ω is anabsolute angular velocity of the flexible spacecraft, ω^(x) indicates across product matrix, {dot over (ω)} is a derivative of the absoluteangular velocity ω of the flexible spacecraft, δ is a rigid-flexiblecoupling matrix, η is a modal coordinate, {umlaut over (η)} is asecond-order derivative of the modal coordinate η; u is control input, dis environmental disturbance, C is the damping matrix of the flexibleappendage, and is expressed as: C=diag{2ξ_(i)ω_(ni),i=1, 2, . . . ,n}εR^(n×n); D is the rigidity matrix of the flexible appendage, and isexpressed as: D=diag{ω_(ni) ²,i=1, 2, . . . , n}εR^(n×n), in which ξ_(i)is a damping coefficient, ω_(ni) is a natural frequency, and n is modalnumber.

A spacecraft attitude control system Σ₁ is converted into a spacecraftattitude control system Σ₂ through a mathematical conversion. Thespacecraft attitude control system Σ₂ is incorporated into the sum offlexible vibration and environmental disturbance. The spacecraftattitude control system Σ₂ is expresses as: Σ₂: J₀{dot over(ω)}=−ω^(x)Jω+u+d, in which, a coefficient matrix J₀=J−δ^(T)δ, and thesum d of flexible vibration and external environmental disturbance isexpresses as: d=δ^(T)(C{dot over (η)}+Dη)+d.

In step S102, describe the sum of flexible vibration and externalenvironmental disturbance by constructing an external system Σ₃.

Firstly, an uncertain portion C_(Δ) of a damping matrix C of theflexible appendage of the spacecraft is incorporated, and an uncertainportion D_(Δ) of a rigidity matrix D of the flexible appendage of thespacecraft is incorporated; the damping matrix and rigidity matrix ofthe spacecraft are described as below:

$\quad\left\{ \begin{matrix}{C = {C_{0} + C_{\Delta}}} \\{D = {D_{0} + D_{\Delta}}}\end{matrix} \right.$

in which, C₀ and D₀ are respectively nominal parameters measured on theground.

Secondly, state variables w₁=η, w₂={dot over (η)} and w₃=d are defined,obtaining the following equation:

$\begin{bmatrix}{\overset{.}{w}}_{1} \\{\overset{.}{w}}_{2} \\{\overset{.}{w}}_{3}\end{bmatrix} = {{\begin{bmatrix}0 & I & 0 \\{- {GD}} & {- {GC}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}\begin{bmatrix}w_{1} \\w_{2} \\w_{3}\end{bmatrix}} - {\begin{bmatrix}0 \\G \\0\end{bmatrix}\delta\;{J^{- 1}\left( {{{- \omega^{x}}J\;\omega} + u} \right)}} + \begin{bmatrix}0 \\0 \\\overset{.}{d}\end{bmatrix}}$

in which, I is a unit matrix, and a matrix G=(I−δJ⁻¹δ^(T))⁻¹; acoefficient matrix is defined:

${W = \begin{bmatrix}0 & I & 0 \\{- {GD}_{0}} & {- {GC}_{0}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}},{B = {{- \begin{bmatrix}0 \\G \\0\end{bmatrix}}\delta\; J^{- 1}}},{{{{and}\mspace{14mu} V} = \left\lbrack {\delta^{T}D\mspace{14mu}\delta^{T}C\mspace{14mu} I} \right\rbrack};}$

Finally, an external system Σ₃ is constructed and is described as below:

$\sum\limits_{3}{:\left\{ \begin{matrix}{\overset{.}{w} = {{\left( {W + W_{\Delta}} \right)w} + {B\left( {{{- \omega^{x}}J\;\omega} + u} \right)} + \Gamma}} \\{\overset{\_}{d} = {V\; w}}\end{matrix} \right.}$in which, w=[w₁ ^(T) w₂ ^(T) w₃ ^(T)]^(T), Γ is an uncertain vector, andΓ is expressed as: Γ=[0 0 {dot over (d)}]^(T); W_(Δ) satisfies a boundedcondition W_(Δ)=MF(t)N, M and N are constant matrixes of a proper numberof dimensions, F(t) is a time-varying matrix and satisfiesF^(T)(t)F(t)≦I; a state variable w satisfies a norm bounded condition∥w∥≦α, the sum d of flexible vibration and environmental disturbancesatisfies a norm bounded condition ∥d∥≦β, in which α and β are knownconstants.

In step S103, configuring a sliding mode disturbance observer forestimating the value of the sum of flexible vibration and environmentaldisturbance.

The sum of flexible vibration and external environmental disturbance isincorporated into the spacecraft attitude control system built in stepS101, and it needs to estimate the value of the sum of flexiblevibration and external environmental disturbance. In the embodiment ofthe present invention, specifically, a sliding mode disturbance observeris used to estimate the sum of flexible vibration and externalenvironmental disturbance.

The steps for configuring the sliding mode disturbance observer are asfollows:

(1) constructing an auxiliary system Σ₄, which is expressed as: J₀{dotover (ω)}=−ω^(x)Jω+u+v, in which {circumflex over (ω)} is a statevariable of the auxiliary system, and v is a sliding mode term;

(2) making the sliding mode term

${v = {k\frac{\overset{\sim}{\omega}}{\omega }}},$in which, {tilde over (ω)}=ω−{circumflex over (ω)}, k>β is a givenconstant, and converting the auxiliary system Σ₄ to a system Σ₅, whichis expressed as: J₀{tilde over ({dot over (ω)})}=d−v;

(3) constructing a Lyapunov function V₁={tilde over (ω)}^(T)J₀{tildeover (ω)} for the system Σ₅, and taking derivative for the Lyapunovfunction V₁, obtaining the following relationship:

${{\overset{.}{V}}_{1} \leq {{- 2}\frac{\left( {k - \beta} \right)}{\sqrt{\lambda_{m\; a\; x}\left( J_{0} \right)}}V_{1}^{\frac{1}{2}}}},$in which, λ_(max)(J₀) is the maximum eigenvalue of J₀; then {tilde over(ω)} is converged to zero within a limited time of t_(r), the slidingmode term v is equivalent to d, wherein

${t_{r} = \frac{\sqrt{\lambda_{m\; a\; x}\left( J_{0} \right)}{V_{1}^{\frac{1}{2}}(0)}}{k - \beta}},$V₁(0) is an initial value of the Lyapunov function V₁.

(4) configuring a sliding mode disturbance observer Σ₆, which isexpressed as:

$\sum\limits_{6}{:\left\{ \begin{matrix}{{\hat{\overset{\_}{d}} = {V\;\hat{w}}},{\hat{w} = {\xi + {L\; J_{0}\omega}}}} \\{\overset{.}{\xi} = {{\left( {W - {LV}} \right)\hat{w}} + {\left( {B - L} \right)\left( {{{- \omega^{x}}J\;\omega} + u} \right)} + {\gamma\; P^{- 1}V^{T}{{sign}(v)}}}}\end{matrix} \right.}$

in which, {circumflex over (d)} is the estimated value of the sum d offlexible vibration and environmental disturbance, ŵ is an estimatedvalue of the state variable w, ξ is an auxiliary state variable, L is anobserver gain to be determined, γ>0 is an adjustable constant, P>0 is apositive definite symmetric matrix to be solved, sign(·) is a signfunction, and for a n-dimensional vector,

x = [x₁  …  x_(n)]^(T),the sign function sign(·) satisfies

sign(x) = [sign(x₁)  …  sign(x_(n))]^(T).

In the present embodiment, the observer gain L to be determined and thepositive definite symmetric matrix P to be solved are solved by way ofan inequality as below:

The positive definite symmetric matrix P and a matrix P_(L) satisfy thefollowing linear matrix inequality:

$\begin{bmatrix}{\left( {{PW} - {P_{L}V}} \right) + \left( {{PW} - {P_{L}V}} \right)^{T}} & {PM} & P \\* & {{- \mu_{1}}I} & 0 \\* & * & {{- \mu_{2}}I}\end{bmatrix} < 0$in which, μ₁>0, μ₂>0 are given constants, I is a unit matrix with aproper number of dimensions, the symbol “*” represents a symmetricalportion of a symmetrical matrix, and the gain matrix is selected to beL=P⁻¹P_(L). The sliding mode disturbance observer constructed in thisembodiment observes that an error e_(w) asymptotically converges into anadjustable area Ω near a balance point, and the adjustable area Ω, whichis expressed as:

$\Omega = \left\{ {e_{w} \in {R^{2\; n}{\left. {{{V\; e_{w}}} \leq \frac{{\mu_{1}\alpha{N}^{2}} + {\mu_{2}{\Gamma }^{2}}}{2\;\gamma}} \right\}.}}} \right.$

A gain array of linear feedback is easily solved using the above linearmatrix inequality in this embodiment. A sliding mode term parameter isselected according to accuracy and rate requirement. Through theselection of the sliding mode term parameter, the observation error isconverged into an adjustable area including an original point, therebyimproving the estimation accuracy of the observer.

In step S104, compounding a nominal controller with the sliding modedisturbance observer to obtain a compound controller stabilizing systemfor compensating for the sum of flexible vibration and environmentaldisturbance.

In the embodiment of the present invention, the sum of flexiblevibration and environmental disturbance is incorporated into aspacecraft attitude control system, and the value of the sum isestimated through a configured disturbance observer. In the embodiment,the spacecraft attitude control system needs to compensate for theestimated value of the sum of flexible vibration and environmentaldisturbance, thus ensuring the precise control of the spacecraftattitude. A compound controller is used to compensate for the sum offlexible vibration and environmental disturbance in the presentinvention.

The nominal controller is compounded with the sliding mode disturbanceobserver configured in step S103 to obtain a compound controller forstabilizing the system. The compound controller stabilizing system isspecifically expressed as: u=u_(n)−{circumflex over (d)}, in which u_(n)is the nominal controller used for stabilizing a nominal system withoutflexible vibration or environmental disturbance, and {circumflex over(d)} is a value of the sum d of flexible vibration and environmentaldisturbance estimated by the sliding mode disturbance observer. In thecompound controller stabilizing system, the control input u issubtracted by the estimated value {circumflex over (d)} of the sum d offlexible vibration and environmental disturbance estimated by thesliding mode disturbance observer on the basis of the nominal controlleru_(n), thereby allowing the compound controller to compensate for thesum d of flexible vibration and environmental disturbance with theestimated value {circumflex over (d)} of the sum of flexible vibrationand environmental disturbance.

In certain aspects, the present invention relates to a spacecraft usingthe method as described above.

By combining the description and practice of the present inventiondisclose herein, other embodiments of the present invention are alsoeasy to conceive and understand for a person skilled in the art. Thedescription and embodiments are only illustrative, and the real scopeand essence of the present invention shall be defined by the claims.

According to the present invention, a sliding mode disturbance observerwith high observation accuracy, strong robustness and easy gainscheduling is provided, which solves the difficulty of accurateestimation and compensation for disturbance with uncertain parametersthat can be modeled, and improves the control accuracy of the system.

Based on the description and practice of the present invention asdisclosed herein, other embodiments of the present invention are readilyconceived of and understood to those skilled in the art. The descriptionand embodiments are provided for exemplary purpose only, the real scopeand spirit of the present invention are defined by the claims.

Other embodiments will be conceivable and understood by those skilled inthe art upon consideration of this description or from practice of theinvention disclosed herein. The description and embodiments are merelyexemplary, and the true scope and spirit are intended to be defined bythe claims.

What is claimed is:
 1. A method for disturbance compensation based on asliding mode disturbance observer for a spacecraft with a large flexibleappendage, comprising the following steps of: a) building a spacecraftattitude control system Σ₁, the spacecraft attitude control system Σ₁being incorporated with environmental disturbance and being convertedinto the spacecraft attitude control system Σ₂, and the spacecraftattitude control system Σ₂ being incorporated with a sum of flexiblevibration and environmental disturbance; b) constructing an externalsystem Σ₃, the external system Σ₃ describing the sum of flexiblevibration and environmental disturbance; wherein, the external system Σ₃is constructed through the following steps: (1) incorporating anuncertain portion C_(Δ) of a damping matrix C of the flexible appendageof the spacecraft, and incorporating an uncertain portion D_(Δ) of arigidity matrix D of the flexible appendage of the spacecraft;describing the damping matrix and rigidity matrix of the spacecraft asbelow: $\quad\left\{ \begin{matrix}{C = {C_{0} + C_{\Delta}}} \\{D = {D_{0} + D_{\Delta}}}\end{matrix} \right.$ in which, C₀ and D₀ are respectively nominalparameters measured on the ground; (2) defining state variables w₁=η,w₂={dot over (η)} and w₃=d, obtaining the following equation:$\begin{bmatrix}{\overset{.}{w}}_{1} \\{\overset{.}{w}}_{2} \\{\overset{.}{w}}_{3}\end{bmatrix} = {{\begin{bmatrix}0 & I & 0 \\{- {GD}} & {- {GC}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}\begin{bmatrix}w_{1} \\w_{2} \\w_{3}\end{bmatrix}} - {\begin{bmatrix}0 \\G \\0\end{bmatrix}\delta\;{J^{- 1}\left( {{{- \omega^{x}}J\;\omega} + u} \right)}} + \begin{bmatrix}0 \\0 \\\overset{.}{d}\end{bmatrix}}$ in which, I is a unit matrix, and a matrixG=(I−δJ⁻¹δ^(T))⁻¹; (3) defining the following coefficient matrix:${W = \begin{bmatrix}0 & I & 0 \\{- {GD}_{0}} & {- {GC}_{0}} & {{- G}\;\delta\; J^{- 1}} \\0 & 0 & 0\end{bmatrix}},{B = {{- \begin{bmatrix}0 \\G \\0\end{bmatrix}}\delta\; J^{- 1}}},{and}$ V = [δ^(T)D  δ^(T)C  I]; (4) theexternal system Σ₃ being described as below:$\sum\limits_{3}{:\left\{ \begin{matrix}{\overset{.}{w} = {{\left( {W + W_{\Delta}} \right)w} + {B\left( {{{- \omega^{x}}J\;\omega} + u} \right)} + \Gamma}} \\{\overset{\_}{d} = {V\; w}}\end{matrix} \right.}$ in which, w=[w₁ ^(T) w₂ ^(T) w₃ ^(T)]^(T), Γ isan uncertain vector, and Γ is expressed as: Γ=[0 0 {dot over (d)}]^(T);W_(Δ) satisfies a bounded condition W_(Δ)=MF(t)N, M and N are constantmatrixes of a proper number of dimensions, F(t) is a time-varying matrixand satisfies F^(T)(t)F(t)≦I; a state variable w satisfies a normbounded condition ∥w∥≦α, the sum d of flexible vibration andenvironmental disturbance satisfies a norm bounded condition ∥d∥≦β, inwhich α and β are known constants; c) configuring a sliding modedisturbance observer for estimating the value of the sum of flexiblevibration and environmental disturbance; d) compounding a nominalcontroller with the sliding mode disturbance observer in step c) toobtain a compound controller; the compound controller compensating forthe sum of flexible vibration and environmental disturbance according tothe estimated value of the sum of flexible vibration and environmentaldisturbance.
 2. The method according to claim 1, wherein the spacecraftattitude control system Σ₁ is expressed as:$\sum\limits_{1}{:\left\{ \begin{matrix}{{{J\;\overset{.}{\omega}} + {\delta^{T}\overset{¨}{\eta}}} = {{{- \omega^{x}}J\;\omega} + u + d}} \\{{\overset{¨}{\eta} + {C\overset{.}{\eta}} + {D\;\eta} + {\delta\overset{.}{\omega}}} = 0}\end{matrix} \right.}$ in which, J is an inertia matrix of the flexiblespacecraft, ω is an absolute angular velocity of the flexiblespacecraft, ω^(x) indicates a cross product matrix, {dot over (ω)} is aderivative of the absolute angular velocity ω of the flexiblespacecraft, δ is a rigid-flexible coupling matrix, η is a modalcoordinate, {umlaut over (η)} is a second-order derivative of the modalcoordinate η; u is control input, d is environmental disturbance, C isthe damping matrix of the flexible appendage, and D is the rigiditymatrix of the flexible appendage.
 3. The method according to claim 2,wherein the damping matrix C of the flexible appendage is expressed as:C=diag{2ξ_(i)ω_(ni),i=1, 2, . . . , n}εR^(n×n), and the rigidity matrixD of the flexible appendage is expressed as: D=diag{ω_(ni) ²,i=1, 2, . .. , n}εR^(n×n), in which is a damping coefficient, ω_(ni) is a naturalfrequency, and n is modal number.
 4. The method according to claim 2,wherein the system Σ₂ is expressed as:Σ₂ : J ₀{dot over (ω)}=−ω^(x) Jω+u+d in which, a coefficient matrixJ₀=J−δ^(T)δ, and the sum d of flexible vibration and environmentaldisturbance is expressed as d=δ^(T)(C{dot over (η)}+Dη)+d.
 5. The methodaccording to claim 1, wherein the steps for configuring the sliding modedisturbance observer are as follows: (1) constructing an auxiliarysystem Σ₄, which is expressed as: J₀{circumflex over ({dot over(ω)})}=−ω^(x)+u+v, in which {circumflex over (ω)} is a state variable ofthe auxiliary system, and v is a sliding mode term; (2) making thesliding mode term ${v = {k\frac{\overset{\sim}{\omega}}{\omega }}},$in which, {tilde over (ω)}=ω−{circumflex over (ω)}, k>β is a givenconstant, and converting the auxiliary system Σ₄ to a system Σ₅, whichis expressed as: J₀{tilde over ({dot over (ω)})}=d−v; (3) constructing aLyapunov function V₁={tilde over (ω)}^(T)J₀{tilde over (ω)} for thesystem Σ₅, and taking derivative for the Lyapunov function V₁, obtainingthe following relationship:${{\overset{.}{V}}_{1} \leq {{- 2}\frac{\left( {k - \beta} \right)}{\sqrt{\lambda_{m\; a\; x}\left( J_{0} \right)}}V_{1}^{\frac{1}{2}}}},$in which, λ_(max)(J₀) is the maximum eigenvalue of J₀; (4) configuring asliding mode disturbance observer Σ₆, which is expressed as:$\sum\limits_{6}{:\left\{ \begin{matrix}{{\hat{\overset{\_}{d}} = {V\;\hat{w}}},{\hat{w} = {\xi + {L\; J_{0}\omega}}}} \\{\overset{.}{\xi} = {{\left( {W - {LV}} \right)\hat{w}} + {\left( {B - L} \right)\left( {{{- \omega^{x}}J\;\omega} + u} \right)} + {\gamma\; P^{- 1}V^{T}{{sign}(v)}}}}\end{matrix} \right.}$ in which, {circumflex over (d)} is the estimatedvalue of the sum d of flexible vibration and environmental disturbance,ŵ is an estimated value of the state variable w, ξ is an auxiliary statevariable, L is an observer gain to be determined, γ>0 is an adjustableconstant, P>0 is a positive definite symmetric matrix to be solved,sign(·) is a sign function, and for a n-dimensional vector,x = [x₁  …  x_(n)]^(T), the sign function sign(·) satisfiessign(x) = [sign(x₁)  …  sign(x_(n))]^(T).
 6. The compensation methodaccording to claim 5, wherein in the step (3), {tilde over (ω)} isconverged to zero within a limited time of t_(r), the sliding mode termv is equivalent to d, wherein${t_{r} = \frac{\sqrt{\lambda_{m\; a\; x}\left( J_{0} \right)}{V_{1}^{\frac{1}{2}}(0)}}{k - \beta}},$V₁(0) is an initial value of the Lyapunov function V₁.
 7. Thecompensation method according to claim 1, wherein, the sliding modedisturbance observer observes that an error e_(w) asymptoticallyconverges into an adjustable area Ω near a balance point, and theadjustable area Ω is expressed as:$\Omega = \left\{ {{e_{w} \in {R^{2\; n}\left. {{{Ve}_{w}} \leq \frac{{\mu_{1}\alpha{N}^{2}} + {\mu_{2}{\Gamma }^{2}}}{2\;\gamma}} \right\}}},} \right.$wherein, μ₁>0, μ₂>0 are given constants.
 8. The compensation methodaccording to claim 5, wherein, the observer gain L to be determined andthe positive definite symmetric matrix P to be solved are solved asbelow: the positive definite symmetric matrix P and a matrix P_(L)satisfy the following linear matrix inequality: $\begin{bmatrix}{\left( {{PW} - {P_{L}V}} \right) + \left( {{PW} - {P_{L}V}} \right)^{T}} & {PM} & P \\* & {{- \mu_{1}}I} & 0 \\* & * & {{- \mu_{2}}I}\end{bmatrix} < 0$ in which, μ₁>0, μ₂>0 are given constants, I is a unitmatrix with a proper number of dimensions, the symbol “*” represents asymmetrical portion of a symmetrical matrix, and the gain matrix isselected to be L=P⁻¹P_(L).
 9. The compensation method according to claim1, wherein, the compound controller is expressed as: u=u_(n)−{circumflexover (d)}, u_(n) is the nominal controller used for stabilizing anominal system without flexible vibration or environmental disturbance,and {circumflex over (d)} is a value of the sum d of flexible vibrationand environmental disturbance estimated by the sliding mode disturbanceobserver.